# How to Calculate Uncertainty

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Quantifying the level of uncertainty in your measurements is a crucial part of science. No measurement can be perfect, and understanding the limitations on the precision in your measurements helps to ensure that you don’t draw unwarranted conclusions on the basis of them. The basics of determining uncertainty are quite simple, but combining two uncertain numbers gets more complicated. The good news is that there are many simple rules you can follow to adjust your uncertainties regardless of what calculations you do with the original numbers.

#### TL;DR (Too Long; Didn't Read)

If you’re adding or subtracting quantities with uncertainties, you add the absolute uncertainties. If you’re multiplying or dividing, you add the relative uncertainties. If you’re multiplying by a constant factor, you multiply absolute uncertainties by the same factor, or do nothing to relative uncertainties. If you’re taking the power of a number with an uncertainty, you multiply the relative uncertainty by the number in the power.

## Estimating the Uncertainty in Measurements

Before you combine or do anything with your uncertainty, you have to determine the uncertainty in your original measurement. This often involves some subjective judgment. For example, if you’re measuring the diameter of a ball with a ruler, you need to think about how precisely you can really read the measurement. Are you confident you’re measuring from the edge of the ball? How precisely can you read the ruler? These are the types of questions you have to ask when estimating uncertainties.

In some cases you can easily estimate the uncertainty. For example, if you weigh something on a scale that measures down to the nearest 0.1 g, then you can confidently estimate that there is a ±0.05 g uncertainty in the measurement. This is because a 1.0 g measurement could really be anything from 0.95 g (rounded up) to just under 1.05 g (rounded down). In other cases, you’ll have to estimate it as well as possible on the basis of several factors.

#### Tips

• Significant Figures: Generally, absolute uncertainties are only quoted to one significant figure, apart from occasionally when the first figure is 1. Because of the meaning of an uncertainty, it doesn’t make sense to quote your estimate to more precision than your uncertainty. For instance, a measurement of 1.543 ± 0.02 m doesn’t make any sense, because you aren’t sure of the second decimal place, so the third is essentially meaningless. The correct result to quote is 1.54 m ± 0.02 m.

## Absolute vs. Relative Uncertainties

Quoting your uncertainty in the units of the original measurement – for example, 1.2 ± 0.1 g or 3.4 ± 0.2 cm – gives the “absolute” uncertainty. In other words, it explicitly tells you the amount by which the original measurement could be incorrect. The relative uncertainty gives the uncertainty as a percentage of the original value. Work this out with:

\text{Relative uncertainty} = \frac{\text{absolute uncertainty}}{\text{best estimate}} × 100\%

So in the example above:

\text{Relative uncertainty} = \frac{0.2 \text{ cm}}{3.4\text{ cm}} × 100\% = 5.9\%

The value can therefore be quoted as 3.4 cm ± 5.9%.

Work out the total uncertainty when you add or subtract two quantities with their own uncertainties by adding the absolute uncertainties. For example:

(3.4 ± 0.2 \text{ cm}) + (2.1 ± 0.1 \text{ cm}) = (3.4 + 2.1) ± (0.2 + 0.1) \text{ cm} = 5.5 ± 0.3 \text{ cm} \\ (3.4 ± 0.2 \text{ cm}) - (2.1 ± 0.1 \text{ cm}) = (3.4 - 2.1) ± (0.2 + 0.1) \text{ cm} = 1.3 ± 0.3 \text{ cm}

## Multiplying or Dividing Uncertainties

When multiplying or dividing quantities with uncertainties, you add the relative uncertainties together. For example:

(3.4 \text{ cm} ± 5.9\%) × (1.5 \text{ cm} ± 4.1\%) = (3.4 × 1.5) \text{ cm}^2 ± (5.9 + 4.1)\% = 5.1 \text{ cm}^2 ± 10\%
\frac{(3.4 \text{ cm} ± 5.9\%)}{(1.7 \text{ cm} ± 4.1 \%)} = \frac{3.4}{1.7} ± (5.9 + 4.1)\% = 2.0 ± 10%

## Multiplying by a Constant

If you’re multiplying a number with an uncertainty by a constant factor, the rule varies depending on the type of uncertainty. If you’re using a relative uncertainty, this stays the same:

(3.4 \text{ cm} ± 5.9\%) × 2 = 6.8 \text{ cm} ± 5.9\%

If you’re using absolute uncertainties, you multiply the uncertainty by the same factor:

(3.4 ± 0.2 \text{ cm}) × 2 = (3.4 × 2) ± (0.2 × 2) \text{ cm} = 6.8 ± 0.4 \text{ cm}

## A Power of an Uncertainty

If you’re taking a power of a value with an uncertainty, you multiply the relative uncertainty by the number in the power. For example:

(5 \text{ cm} ± 5\%)^2 = (5^2 ± [2 × 5\%]) \text{ cm}^2 = 25 \text{ cm}^2± 10\% \\ \text{Or} \\ (10 \text{ m} ± 3\%)^3 = 1,000 \text{ m}^3 ± (3 × 3\%) = 1,000 \text{ m}^3 ± 9\%

You follow the same rule for fractional powers.

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